The Mountain Pass Theorem and Applications

Abstract

This project lies at the interface between Nonlinear Functional Analysis, unconstrained Optimization and Critical point theory. It concerns mainly the Ambrosetti-Rabinowitz's Mountain Pass Theorem which is a min-max theorem at the heart of deep mathematics and plays a crucial role in solving many variational problems. As application, a model of Lane-Emden equation is considered. Minmax theorems characterize a critical value c of a functional f defi ned on a Banach spaces as minmax over a suitable class subsets of X, A f of that is : c = inf sup f (x) . A∈S x∈A Variational methods refer to proofs established by showing that a suitable auxilliary function attains a minimum or has a critical point (see below). Minimum Variational principle can be viewed as a mathematical form of the principle of least action in Physics and justi fies why so many results in Mathematics are related to variational techniques since they have their origin in the physical sciences. The application of the Mountain Pass Theorem and more generally those of Variational Techniques cover numerous theoretic as well as applied areas of mathematical sciences such as Partial Differential Equations, Optimization, Banach space geometry, Control theory, Economics and Game theory.